The thermoelectric electromotive force $e$ is given by $e = \alpha t - \frac{1}{2}\beta t^2$. If the temperature of the cold junction is $0 \, ^\circ C$,find the temperature of inversion $t_i$. (Given: $\alpha = 500.0 \, \mu V/^\circ C$,$\beta = 5.0 \, \mu V/^\circ C^2$)

In the circuit shown in the figure,the capacitor $C$ is initially uncharged and the key $K$ is open. In this condition,a current of $1 \,A$ flows through the $1 \,\Omega$ resistor. The key is closed at time $t=t_0$. Which of the following statement(s) is(are) correct?

[Given: $e^{-1}=0.36$]
$(A)$ The value of the resistance $R$ is $3 \,\Omega$.
$(B)$ The current through the $3 \,\Omega$ resistor (connected in parallel to the $1 \,\Omega$ and $R$ branches) is $2 \,A$ when $K$ is open.
$(C)$ At $t=t_0+7.2 \,\mu s$,the current in the capacitor branch is $0.6 \,A$.
$(D)$ For $t < \infty$,the charge on the capacitor is $12 \,\mu C$.

Two different metals are joined end to end. One end is kept at a constant temperature and the other end is heated to a very high temperature. The graph depicting the thermo $e.m.f.$ $(E)$ versus temperature $(t)$ is:

Consider four circuits shown in the figure below. In which circuit is the power dissipated the greatest? (Neglect the internal resistance of the power supply)

$A$ battery of $emf$ $E$ and internal resistance $r$ is connected across a resistance $R$. Resistance $R$ can be adjusted to any value greater than or equal to zero. $A$ graph is plotted between the current $(i)$ passing through the resistance and potential difference $(V)$ across it. Select the correct alternative$(s)$.

For the circuit shown in the figure:
$(A)$ The current $I$ through the battery is $7.5 \text{ mA}$.
$(B)$ The potential difference across $R_L$ is $18 \text{ V}$.
$(C)$ The ratio of powers dissipated in $R_1$ and $R_2$ is $3$.
$(D)$ If $R_1$ and $R_2$ are interchanged,the magnitude of the power dissipated in $R_L$ will decrease by a factor of $9$.